EP1527521B1 - Device and method for robust decoding of arithmetic codes - Google Patents

Description

The invention relates to the compression / decompression of digital data, especially for multimedia signals (audio, image, video, speech), and the robust transmission of these data over noisy networks, such as wireless networks, mobile communications.

To reduce the transmission rate of digital data, they are compressed, seeking to approach the theoretical maximum that specialists call "the entropy of the signal". For this, one often uses statistical codes also called variable length codes, for example the codes of Huffman. These codes nevertheless have the disadvantage of being very sensitive to transmission errors. Inversion of a bit can lead to desynchronization of the decoder, which results in erroneous decoding of all the data following the position of the erroneous bit.

Current solutions for compression, transmission and decompression of network multimedia signals, to which we will return, are based on the assumption that a certain quality of service of data transport is guaranteed. In other words, they assume that the transport and link layers, relying on the use of correcting codes, will make it possible to achieve a residual error rate that is almost zero (ie seen by the application of compression and decompression). But this assumption of almost zero residual error rate is no longer true when the characteristics of the channels vary over time (non-stationary channels), especially in wireless and mobile networks, and for a complexity of the realistic channel code. In addition, the addition of redundancy by error correction codes leads to a reduction in the payload.

The document Fabre et al. "Joint source-channel turbo decoding of VLC-coded Markovsources", 2001 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING. PROCEEDINGS (CAT.NO.01CH37221), 2001 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING. PROCEEDINGS, SALT LAKE CITY, UT, USA, 7-11 MAY 2001, 2657-2660 vol.4, XP002267571, 2001, Piscataway, NJ, USA, IEEE, USA, ISBN: 0-7803-7041-4 describes an iterative method of source-channel joint decoding inspired by the turbo codes in which the three models of the coding chain, ie a model of the source of symbols, a model of the source coder and a model of the channel coder, are used alternately.

The need is therefore felt for new coding / decoding solutions.

The present invention proposes advances in this direction.

• recevoir dudit canal un flux de N seconds éléments binaires Yi correspondants à N premiers éléments binaires Ui transmis, par l'intermédiaire dudit canal, d'un codeur arithmétique ou quasi-arithmétique vers ledit décodeur avec i compris entre 1 et N, ledit codeur codant une source de symboles;
• décoder lesdits N seconds éléments binaires Yi en un flux de K symboles x-aires Sj avec j compris entre 1 et K.

For this purpose, the invention relates to a method of decoding, in an arithmetic or quasi-arithmetic decoder, bits received from a channel in x-ary symbols comprising the following steps:

• receiving from said channel a stream of N second bits Yi corresponding to N first bits Ui transmitted, via said channel, an arithmetic or quasi-arithmetic coder to said decoder with i between 1 and N, said coder coding a source of symbols;
• decoding said N second bits Yi into a stream of K x-area symbols Sj with j between 1 and K.

1. a. calculer des probabilités d'aboutir dans des états produits connaissant des bits du flux de seconds éléments binaires Yi, à partir d'un modèle dudit canal, d'un modèle de ladite source et d'un modèle produit définissant une correspondance entre lesdits symboles x-aires Sj et lesdits premiers éléments binaires Ui, à partir d'états produits dans un diagramme d'états "produit", un état produit comprenant un état source et un état d'un modèle dudit codeur ou dudit décodeur arithmétique ou quasi-arithmétique, et
2. b. déterminer le flux de symboles x-aires le plus probable à partir des probabilités calculées à l'étape a.

Advantageously, the decoding step comprises:

1. at. calculating probabilities of ending in produced states knowing bits of the stream of second bits Yi, from a model of said channel, a model of said source and a product model defining a correspondence between said symbols x Sj and said first bits Ui, from states produced in a "product" state diagram, a product state comprising a source state and a state of a model of said encoder or said arithmetic or quasi-arithmetic decoder , and
2. b. determine the most likely x-ary symbol flow from the probabilities calculated in step a.

The invention also relates to an arithmetic or quasi-arithmetic decoding device corresponding to the method.

• la figure 1 illustre un ensemble comprenant un codeur et un décodeur selon l'invention,
• la figure 2 est une vue générale du procédé de codage implémentable dans un codeur selon l'invention,
• la figure 3 est un diagramme formant vue générale du procédé de décodage implémentable dans un décodeur selon l'invention,
• les figures 4A, 4B et 4C illustrent trois représentations différentes d'un même modèle dit "source" établissant une correspondance entre les symboles de la source et des séquences de bits,
• les figures 5A et 5B illustrent deux représentations différentes d'une même correspondance entre des séquences de symboles et des séquences de bits, basées sur le modèle dit "source" des figures 4A et 4B,
• la table 1 illustre une représentation sous forme de table d'un modèle dit "codeur",
• la table 2 illustre une représentation sous forme de table d'un modèle dit "décodeur",
• la figure 6A illustre une représentation sous forme de table d'un modèle dit "produit source-codeur"selon l'invention,
• la figure 6B illustre une représentation sous forme de treillis du modèle dit "produit source-codeur" de la figure 6A selon l'invention,
• les figures 7A et 7B illustrent deux représentations partielles sous forme de treillis du modèle dit "codeur" de la table 1,
• la figure 8 illustre une représentation généralisée d'un modèle dit "produit source-codeur"selon l'invention,
• la figure 9A illustre une représentation sous forme de table d'un modèle dit "produit source-décodeur"selon l'invention,
• la figure 9B illustre une représentation sous forme de treillis du modèle dit "produit source-décodeur" de la figure 9A selon l'invention,
• la figure 10 illustre une représentation généralisée d'un modèle dit "produit source-décodeur"selon l'invention,
• la figure 11 illustre le procédé de décodage dans une première réalisation selon l'invention,
• la figure 12 illustre le procédé de décodage dans une deuxième réalisation selon l'invention,
• la figure 13 illustre une représentation sous forme de treillis réalisée à partir du procédé de décodage de la figure 11 ou 12 ,
• la figure 14 illustre le procédé de décodage dans une troisième réalisation selon l'invention,
• la figure 15 illustre le procédé de décodage dans une quatrième réalisation selon l'invention.

Other features and advantages of the invention will appear on examining the detailed description below, and the attached drawings, in which:

• the figure 1 illustrates an assembly comprising an encoder and a decoder according to the invention,
• the figure 2 is a general view of the coding method that can be implemented in an encoder according to the invention,
• the figure 3 is a diagram forming a general view of the decoding method that can be implemented in a decoder according to the invention,
• the Figures 4A, 4B and 4C illustrate three different representations of the same model called "source" establishing a correspondence between the symbols of the source and bit sequences,
• the Figures 5A and 5B illustrate two different representations of the same correspondence between symbol sequences and bit sequences, based on the so-called "source" model of Figures 4A and 4B ,
• Table 1 illustrates a table representation of a so-called "encoder" model,
• Table 2 illustrates a table representation of a so-called "decoder" model,
• the Figure 6A illustrates a table representation of a so-called "source-encoder product" model according to the invention,
• the Figure 6B illustrates a lattice representation of the so-called "source-encoder product" model of the Figure 6A according to the invention,
• the Figures 7A and 7B illustrate two partial representations in the form of lattices of the so-called "encoder" model of table 1,
• the figure 8 illustrates a generalized representation of a so-called "source-encoder product" model according to the invention,
• the Figure 9A illustrates a table representation of a so-called "source-decoder product" model according to the invention,
• the Figure 9B illustrates a lattice representation of the so-called "source-decoder product" model of the Figure 9A according to the invention,
• the figure 10 illustrates a generalized representation of a so-called "source-decoder product" model according to the invention,
• the figure 11 illustrates the decoding method in a first embodiment according to the invention,
• the figure 12 illustrates the decoding method in a second embodiment according to the invention,
• the figure 13 illustrates a lattice representation made from the decoding method of the figure 11 or 12 ,
• the figure 14 illustrates the decoding method in a third embodiment according to the invention,
• the figure 15 illustrates the decoding method in a fourth embodiment according to the invention.

• l'annexe 1 contient des expressions mathématiques utilisées dans la présente description.

In addition:

• Appendix 1 contains mathematical expressions used in this description.

The drawings and appendices to the description include, for the most part, elements of a certain character. They can therefore not only serve to better understand the description, but also contribute to the definition of the invention, if any.

In general, a multimedia signal compression system (image, video, audio, speech) uses statistical codes also called variable length codes. These allow to obtain speeds approaching what the specialists call "the entropy of the signal". The most used codes in existing systems (especially in standards) are the Huffman codes that have been described in the following book: DA Huffman: "A method for the construction of minimum redundancy codes", Proc. IRE, 40 (1951), p.1098-1101 .

1. [1]- J.J. Rissanen "Generalized kraft inequality and arithmetic", IBM J.Res. Develop., 20:198-203, Mai 1976
2. [2]- J.J. Rissanen, "Arithmetic Coding as number representations", Acta Polytech. Scand. Math., 31:44-51, Décembre 1979

ainsi que dans des brevets américains US 4 286 256 , US 4 467 317 , US 4 652 856 .More recently, there has been a resurgence of interest in arithmetic codes because of their increased performance in terms of compression. They make it possible to decouple the encoding process from the assumed model of the source. This makes it easy to use higher order statistical models. These arithmetic codes have been described in books such as

1. [1] - JJ Rissanen "Generalized kraft inequality and arithmetic", IBM J.Res. Develop., 20: 198-203, May 1976
2. [2] - JJ Rissanen, "Arithmetic Coding and Number Representations", Acta Polytech. Scand. Math. 31: 44-51, December 1979

as well as in US patents US 4,286,256 , US 4,467,317 , US 4,652,856 .

Until recently, the design of the compression systems was done assuming a quality of service transport guaranteed. In fact, it was assumed that the lower layers of the OSI model incorporate error-correcting codes that guarantee a residual error rate given the quasi-zero application.

Codes of varying lengths could therefore be widely used despite their high sensitivity to transmission noise. This assumption of almost zero residual error rate is no longer true in wireless and mobile networks, whose channel characteristics vary over time (non-stationary channels). This residual error rate seen by the decoder of the source signals is often far from negligible. Any error in the bit stream may cause the decoder to de-synchronize and thus propagate errors over the sequence of decoded information.

To overcome this propagation problem, the first generation standards (H.261, H.263, MPEG-1, MPEG-2) have incorporated synchronization markers into the syntax of the transmitted bitstream. These are long code words (16 or 22 bits consisting of a sequence of 15 or 21 bits to '1' followed by a '0') not emulable by errors occurring on the other code words and which can therefore be recognized by the decoder with a probability close to '1'.

This leads to structuring the bitstream into packets delimited by these synchronization markers. This helps to confine the propagation of errors within the packet.

However, if an error occurs at the beginning of the packet, the rest of the packet may be lost. In addition, the periodicity of these synchronization markers must be restricted to avoid excessive loss of compression efficiency.

The new standards (H.263 + and MPEG-4) then used reversible variable length codes (RVLC). The peculiarity of these codes is that they can be decoded from the first to the last bit of a packet, and, conversely, from the last to the first bit of the packet.

If an error occurred in the middle of the packet, this symmetry of the code makes it possible to confine the propagation of errors on a segment in the middle of the packet instead of propagating it until the end of the packet delimited by a synchronization marker. However, the symmetry of the code generates a loss in compression efficiency with respect to a Huffman code of the order of 10%. In addition, reversible variable length codes do not completely avoid the problem of error propagation: if an error occurs at the beginning and at the end of a packet, the whole packet may be wrong.

The design of transmission noise-robust coding / decoding methods based on a family of compression-efficient codes (ie, which allow to approach the entropy of the source), constitutes an important stake, in particular for the future systems of multimedia communication (picture, video, audio, speech) mobile. For these systems, new standards are being studied both within the International Telecommunication Union (ITU) and the International Standard Organization (ISO).

Although standards dominate the telecommunications industry, such a family of codes can also find applications in niche markets using proprietary solutions.

The disclosed invention relates in particular to an encoder / decoder device for a family of codes widely used in compression systems such as arithmetic type codes comprising arithmetic codes and quasi-arithmetic codes.

• [3]- I.H. Witten, R. M. Neal and J. G. Cleary. Arithmetic coding for data compression. Communications of the ACM, 30(6):520-540, Juin 1987 ,
• [4] P. G. Howard and J. S. Vitter, Image and Text Compression, pages 85-112. Kluwer Academic Publisher, 1992 .

The principles of arithmetic and quasi-arithmetic coding are developed below. They are developed in more detail in the book [2] and the following works

• [3] - IH Witten, RM Neal and JG Cleary. Arithmetic coding for data compression. Communications of the ACM, 30 (6): 520-540, June 1987 ,
• [4] PG Howard and JS Vitter, Image and Text Compression, pages 85-112. Kluwer Academic Publisher, 1992 .

In the rest of the description, the use of the word "sequence" designates a stream or a sequence of elements. For example, a sequence of bits or symbols will designate a stream or a sequence of bits, or symbols respectively.

• soit A = A₁ ... A_L, une séquence de symboles de source quantifiés qui prennent leur valeur dans un alphabet fini A composé de M = 2^q symboles, A={a₁ ... a_M}. Cet alphabet a une distribution stationnaire qui indique pour chaque symbole une probabilité stationnaire associée. Cette séquence de symboles M-aires (appelée également source A) est convertie en une séquence de symboles x-aires (avec x≤M). Dans le cas où cette séquence de symboles M-aires est convertie en une séquence de symboles binaires S=S₁...S_K, (appelée également source S), K est définit par K=qxL. Les variables M et x définissent des entiers à partir de 2. Ainsi, dans l'exemple, x = 2 et M=4.

The principle of coding generally consists of the following steps:

• let A = A ₁ ... A _{L be} a sequence of quantized source symbols which take their value in a finite alphabet A composed of M = 2 ^q symbols, A = {a ₁ ... a _M }. This alphabet has a stationary distribution which indicates for each symbol an associated stationary probability. This sequence of M-ary symbols (also called source A) is converted into a sequence of x-ary symbols (with x≤M). In the case where this sequence of M -ary symbols is converted into a sequence of binary symbols S = S ₁ ... S _K , (also called source S), K is defined by K = qxL. The variables M and x define integers starting from 2. Thus, in the example, x = 2 and M = 4.

By way of example only in the remainder of the description, and for reasons of simplification, it will be considered that x = 2. This conversion uses a source model described below comprising a correspondence between the source M-area A and the binary source S. This sequence of binary symbols S is in turn converted into a sequence of first bits of information U = U ₁ ... U _N , via a quasi-arithmetic encoder of precision T. The length N of this bit stream is a random variable, a function of S, the sequence of binary symbols, itself a function of A, the sequence of M -ary symbols. The train of first bits U is transmitted, by way of example only, on a channel without memory. This first bit stream U is received as a stream of second bits Y due to errors introduced during transmission over the communication channel. The lower layers of the communication system make it possible to determine at least the current properties Ic of the communication channel, for example the error rate introduced by the channel into the communicated bits, confidence measurements on bits received. These last measurements make it possible to estimate probabilities of receiving a second binary element knowing a first binary element communicated on the channel. These probabilities are estimated as shown in the appendix (equation 21 with the index i being an integer) from one of the current properties Ic of the channel: p, the probability of error on the channel.

The principle of decoding consists of estimating the sequence of symbols A, knowing the second bits, also called "observations y". In the rest of the description, capital letters will be used to denote random variables, and lowercase letters for realizations, or observations of these variables. To note a sequence of successive variables, the notation X _u ^v = {X _u , X _{u + 1} ... X _v }.

The coding / decoding principle can also be directly applied to a source corresponding not to M-ary symbols (A) with M> 2 but to binary symbols ( S ) with M = 2. In the remainder of the description, an x-ary symbol, respectively an M-ary symbol, comprises in particular the case of the binary symbol (x = 2, respectively M = 2) and the case of the symbol for which x> 2, respectively M > 2.

• un modèle de source 8 permettant de définir des probabilités pour des symboles x-aires associés à des transitions entre états sources dans un diagramme d'états source décrit ci-après,
• un modèle dit "produit" 7 permettant de définir une correspondance entre les symboles x-aires et de premiers éléments binaires, en fonction d'états dans un diagramme d'états "produit", un état produit étant fonction d'un état source et d'un état d'un modèle de transfert.

The figure 1 illustrates an example of a set of encoder 1 and decoder 2. The encoder and the decoder respectively have an input Ec, Ed and an output Wc, Wd. During coding, a stream of x-ary symbols to be coded is received at the input of the encoder in Ec. The encoder transmits at the output Wc a stream of first bits, called the first bit stream Uc, generated by encoding the x-ary symbol stream, for example in compressed form. This first bit stream Uc is sent, via a channel 11, to the input Ed of the decoder 2. The channel, having been able to errone the first bit stream (for example to invert certain bits of the first stream), transmits a stream of second bits, called second bit stream Yd, to the input of the decoder 2. According to the invention, a processing module 6 receives and processes this second bit stream as well as the properties Ic of the channel, communicated via the channel, to obtain at the output Wd of the decoder, a stream of symbols Sd. A processing module 6 is suitable for working with the model module 12. This model module comprising

• a source model 8 making it possible to define probabilities for x-ary symbols associated with transitions between source states in a source state diagram described below,
• a so-called "product" model 7 making it possible to define a correspondence between the x-ary symbols and first binary elements, as a function of states in a "product" state diagram, a product state being a function of a source state and a state of a transfer model.

Product model 7 is derived from source model 8 and transfer model 9 or 10. The product model can be dynamically constructed from the source model and transfer models 9 or 10. In another possible realization , the product model can be implemented in the model module without being built dynamically. The transfer model comprises an encoding model 9 or a decoding model 10. This transfer model makes it possible to establish a correspondence between binary symbols and first bits, as a function of probabilities associated with binary symbols.

It is also possible to start from an M-ary symbol sequence that can be previously transformed into an x-ary symbol sequence before being compressed using the source model.

In one possible embodiment, these models can be pre-calculated and stored in a module external to the decoder model module. They can also be used remotely, for example by using an Ethernet-type local network or a wider Internet-type network.

The processing module 6 is adapted to calculate x-area symbol probabilities, knowing the bits of the second bit stream, from the properties Ic of the channel, the source model and the product model. The processing module 6 is adapted to reconstruct the most likely x-ary symbol stream from these calculated probabilities and to reconstruct the flow of M-ary symbols from the source model if a flow of M-ary symbols has been code.

The method resulting from this device according to the invention is described in the rest of the description and will, in particular, to better understand the operation of such a device.

The figure 2 illustrates an example of a coding method that can use the insertion of synchronization markers. This method is implemented in an encoder receiving as input a source (M-ary), for example a data sequence.

In step 100, when the data sequence does not require M-ary markers, the process proceeds to step 104. If, on the contrary, M-ary markers are to be inserted into the data sequence, the frequency , the size and the value of these markers are provided by the encoder to perform the insertion of these markers in step 102.

In step 104, it is checked whether the data sequence (including or not synchronization markers) requires an M-ary / binary conversion. If this is not the case, then the data sequence is binary and is subject to arithmetic coding (quasi-arithmetic in the example of the figure) in step 112. Otherwise, the M-ary / binary conversion of the sequence is carried out at step 106. In step 108, if bit markers are to be inserted, the frequency, size and value of these markers are provided by the encoder to effect the insertion of these markers at step 110.

In step 112, the M-ary or binary data sequence is subjected to quasi-arithmetic coding whose complexity is controlled by the parameter T. If T is large, the quasi-arithmetic coding becomes identical to an arithmetic coding. This results in a -maximal compression of the bit data sequence into a bit stream.

The figure 3 illustrates an example of a decoding method taking into account the synchronization markers inserted. This method is implemented in a decoder receiving as input the bitstream and information concerning the source M-ary or x-ary (in particular binary), the frequency, the size and the value of the possible synchronization markers.

• un décodage instantané à l'étape 124,
• un décodage séquentiel selon l'invention, qui est détaillé dans le reste de la description, à l'étape 122,
• un décodage optimal, qui est détaillé dans le reste de la description, selon l'invention à l'étape 126.

In step 120, different decoding modes can be selected:

• instant decoding at step 124,
• a sequential decoding according to the invention, which is detailed in the remainder of the description, at step 122,
• an optimal decoding, which is detailed in the rest of the description, according to the invention in step 126.

In step 128, it is checked whether the data sequence obtained after decoding requires a binary / M-ary conversion.

If so, then the data sequence is binary and in step 130, it is checked whether the data sequence comprises binary markers. If so, these markers are removed in step 132 using the information on their frequency and size. After step 130 or step 132, the data sequence is subjected to binary / M-ary conversion in step 134.

After step 128 or 134, it is checked whether the data sequence comprises M-ary markers in step 136. If this is not the case, the decoding of the data (or source) sequence is finished. otherwise, the markers are deleted in step 138 to obtain the data (or source) sequence.

The Figures 4 to 5 illustrate source models for the M-ary / binary and binary / M-ary conversion of a data (or source) sequence.

The Figure 4A illustrates the binary representation of the 4 symbols of a 4-ary source, that is to say a source taking value in a 4-ary alphabet. This source model can be a binary tree of depth q, as shown on the Figure 4A for q = 2 and M = 4. In a variant, this tree can be represented by a correspondence table. The black node is called the root of the tree, the white nodes are the intermediate nodes, the gray nodes are called leaves of the tree. The probabilities P of each M-ary symbol correspond to the stationary probabilities of the source. The transition probabilities (not shown) on the branches of this binary tree are calculated from the stationary probabilities.

The Figure 4B represents the automaton for converting the sequence of independent M-ary symbols into a sequence of binary symbols. This source model can also be represented in Figure 4B in the form of a three-state automaton: C = 0 can be the initial state, C = 1 and C = 2 the intermediate states. Alternatively, this source model can be represented in figure 4C in the form of a lattice with the same states: C = 0 on the left, the initial state connected to the intermediate states C = 1 and C = 2 on the right, these intermediate states, on the left, being connected to the state C = 0 on the right. In addition, in this example, the probabilities of each x-ary symbol (here, binary symbol) are indicated on the corresponding transitions.

For a data flow comprising several non-independent M-ary symbols, a complete model of the source is obtained by connecting successive local models as indicated on the Figure 5A . A local model corresponds to a source model for a flow comprising a symbol as illustrated on the Figure 4A . With reference to figures 5 for a flow of two dependent M-ary symbols, the complete model of the source consists of identifying the leaves of the first binary tree with the root of the next tree, root represented by a black node on the Figures 5A and 5B . The Figure 5A thus combines binary trees between them. Alternatively, the Figure 5B combines a first binary tree with corresponding automata taking as initial and final state the leaves of the binary tree. This automaton makes it possible to convert a sequence of dependent M-ary symbols into a sequence of binary symbols.

According to Figure 5A , C _k states define the possible nodes which can be in the machine after the production of k binary symbols. This set of possible nodes is symbolized in the figure by a mixed line connecting the nodes of the same state k. The sequence C ₀ ,..., C _K is a Markov chain and the flow of corresponding binary symbols is a function of the transitions of this chain. The transitions called S _k since they generate the emission of the binary symbols S _k , represent the set of possible transitions between the set of states C _k-1 and the set of states C _k of the source model.

From this stochastic automaton, the flow of binary symbols can be modeled by a hidden Markov model.

Knowing each stationary probability associated with each binary symbol, each of these probabilities corresponding to one of the segments of the unit interval [0,1 [, the arithmetic coding makes it possible, for a sequence of binary symbols, to determine the interval of this sequence . The latter represents a segment of the unit interval and its lower bound is derived from the sum of the probabilities of the symbol sequences corresponding to the lower subintervals. The width of this interval is given by the probability of the sequence of symbols corresponding to this interval. The "interval current" is the interval corresponding to a part of the sequence and therefore to the state C _k current of this part of sequence, when calculating the interval or sub-segment associated with the sub-sequence corresponding to the state C _k .

• [5]- R. Pasco. Source coding algorithms for fast data compression. PhD thesis, Dept. of Electrical Engineering, Stanford Univ., Stanford Calif.,1976

et revisitées dans [2]. Un problème qui se pose lors de cette mise en oeuvre est la nécessité de disposer d'une grande précision numérique pour représenter des intervalles avec des nombres réels très petits. Cette difficulté peut être résolue en se basant sur la représentation binaire des nombres réels dans l'intervalle [0,1[, ce qui est décrit dans [3].Tout nombre appartenant à l'intervalle [0,0.5[ a son premier bit égal à 0,alors que tout nombre appartenant à l'intervalle [0.5,1[ a son premier bit égal à 1. Par conséquent, pendant le processus de codage, dés que l'intervalle courant est entièrement contenu dans [0,0.5[ ou [0.5,1[, le bit correspondant est émis et la taille de l'intervalle est doublée. Un traitement spécifique est nécessaire pour les intervalles qui chevauchent ½. Si l'intervalle courant chevauche ½ et est entièrement contenu dans [0.25,0.75[, il ne peut pas être identifié par un bit unique. Sa taille est tout de même doublée, sans émettre de bit, et le nombre de fois (n) où cette opération est effectuée avant d'émettre un bit est mémorisée, n est le nombre de remises à l'échelle effectuées depuis la dernière émission d'un bit. Lorsqu'un intervalle pour lequel un bit U_i= u_i est émis est atteint, alors ce bit est suivi par la séquence de n fois le bit de valeur opposée.Practical implementations of arithmetic coding were first introduced in the book [1], the following work:

• [5] - R. Pasco. Source coding algorithms for fast data compression. PhD thesis, Dept. of Electrical Engineering, Stanford University, Stanford Calif., 1976

and revisited in [2]. A problem that arises in this implementation is the need for high numerical precision to represent intervals with very small real numbers. This difficulty can be solved based on the binary representation of the real numbers in the interval [0,1 [, which is described in [3]. Any number belonging to the interval [0,0,5] has its first bit equal to 0, while any number belonging to the interval [0.5,1] has its first bit equal to 1. Therefore, during the coding process, as soon as the current interval is entirely contained in [0,0,5 [ or [0.5,1 [, the corresponding bit is issued and the interval size is doubled. Specific treatment is required for intervals that overlap ½. If the current interval overlaps ½ and is fully contained in [0.25,0.75 [, it can not be identified by a single bit. Its size is still doubled, without emitting a bit, and the number of times (n) where this operation is performed before emitting a bit is stored, n is the number of rescaling done since the last broadcast one bit. When an interval for which a bit U _i = u _i is transmitted is reached, then this bit is followed by the sequence of n times the opposite value bit.

The use of this technique ensures that the current interval continuously satisfies low <0.25 <0.5 ≤ high or low <0.5 <0.75 ≤ high , with low and high the lower and upper limits of the current range respectively.

The principle of arithmetic coding can be modeled by a stochastic automaton whose states E are defined by three variables: low, up and n _scl the number of rescaling. The disadvantage of this coding is that there can be a very large number of states if the stationary distribution of the source is not known a priori or if the stream comprises a large number of x-ary symbols. Quasi-arithmetic coding, also known as low-precision arithmetic coding, reduces the number of possible states without significant loss of compression efficiency, as described in [4]. To carry out the quasi-arithmetic coding, the real interval [0,1 [is replaced by the integer interval [0, T [. The integer T makes it possible to control the compromise between the complexity and the compression efficiency: if T is large, then the subdivisions of the interval will represent precisely the distribution of the source. On the other hand, if T is small (close to 1), all the possible subdivisions of the interval can be pre-calculated, these subdivisions will represent only an approximation of the distribution of the source.

As shown in Table 1, all transitions between states and the corresponding bit transmissions can be pre-calculated. Arithmetic operations are then replaced by table reads.

A sequence S ₁ ^K of binary symbols is converted a bit stream U ₁ ^N by a binary tree. This tree can be seen as an automaton or an encoder model that models the distribution of the bitstream. The encoding of a symbol determines the choice of an arc of the tree, to which the transmission of bits can be associated. Each node corresponds to a state E of the arithmetic coder. The successive transitions between these states follow the distribution of the source P (S _k | S _k-1 ) for a Markov source of order 1. Let E _{k be} the state of the automaton at each "symbol instant" k. As for the arithmetic coding, the state E _k of the quasi-arithmetic encoder is defined by three variables: lowS _k , upS _k and nscl _k . The terms lowS _k and upS _k denote the bounds of the interval resulting from successive subdivisions of [0, T [corresponding to the coding of the sequence S ₁ ^K. The quantity nscl _k is reset every time a bit is issued and incremented each time a rescaling without bit transmission is performed. It therefore represents the number of rescaling done since the last bit transmission. When a bit is sent, it is followed by nscl _k bits of opposite value.

Since there is a finite number of possible subdivisions of the interval [0, T [, it must be possible to pre-calculate all the states of the quasi-arithmetic encoder without knowing the source a priori.

By way of example, Table 1 presents the states, outputs and all possible transitions of a quasi-arithmetic precision encoder T = 4 for a binary source. We will first look at the sets C1 and C2 of columns. The value of the variable nscl _k is not included in the state model. Only the increments of this variable are indicated by the letter f in the table as in the book [4].

• [6] P. G. Howard and J. S. Vitter, Design and analysis of fast text compression based on quasi-arithmetic coding. In Data Compression Conference, pages 98-107, Snowbird, Utah,Mars-Avril 1993 .

pour le choix de ces subdivisions. En fonction de la probabilité du symbole binaire d'entrée 0, l'intervalle courant [lowS_k, upS_k[ sera subdivisé en intervalles entiers. Suivant que, dans la colonne C2, le symbole binaire d'entrée est 0 ou 1, les subdivisions en intervalles entiers de l'intervalle courant [lowS_k, upS_k[ possibles conduisent à l'émission de bits indiquées ou à la remise à l'échelle appropriée avant de déterminer le prochain état de l'automate.The encoder has three possible states, corresponding to integer subdivisions of the interval [0.4 [taken by the current interval [lowS _k , upS _k [. We will refer to the book

• [6] Howard Howard and JS Vitter, Design and analysis of fast compression-based text on quasi-arithmetic coding. In Data Compression Conference, pp. 98-107, Snowbird, Utah, March-April 1993 .

for the selection of these subdivisions. Depending on the probability of the input binary symbol 0, the current interval [lowS _k , upS _k [will be subdivided into integer intervals. According to the fact that, in column C2, the input binary symbol is 0 or 1, the subdivisions into integer intervals of the current interval [lowS _k , upS _k [possible lead to the emission of indicated bits or to the delivery to the appropriate scale before determining the next state of the controller.

In the case where the input symbols are binary symbols, the number of states can be further reduced by identifying the binary symbols as "less likely" (LPS) and "more likely" (MPS) in column C3, rather than 0 and 1 in column C2. This amounts to considering the probability of each binary symbol rather than the value of the binary symbol, the MPS and LPS being able to correspond either to a bit 0 or to a bit 1. The number of possible combinations of the probabilities of the binary source is reduced. This makes it possible to combine certain transitions and to eliminate states of the automaton.

The sequence E ₀ ... E _K of successive states of the quasi-arithmetic encoder forms a Markov chain and the outputs of the encoder are functions of the transitions on this chain.

In the same way as quasi-arithmetic coding, the decoding can be modeled by a stochastic automaton with a number of states E _n finite as presented on the table 2.

The state of an arithmetic decoder is defined by two intervals: [lowU _n , upU _n [and [lowS _Kn , upS _Kn [. The interval [lowU _n , upU _n [corresponds to the segment of the interval [0,1 [defined by the reception of the bit sequence U ₁ ⁿ . The interval [lowS _Kn , upS _Kn [is the segment of the interval [0,1 [obtained when the symbol sequence S ₁ ^Kn has been decoded. This interval is updated by accumulating the probabilities of each decoded binary symbol, in the same way as the coding for each coded symbol. It reproduces the operation of the encoder.

An encoder and an arithmetic decoder are not synchronized. Indeed, in coding, the reception of a symbol does not necessarily lead to the emission of a bit. Similarly, at decoding, reading a bit does not necessarily lead to the production of a symbol. Thus, if the reading of the first symbol of the sequence to be coded causes no bit transmission, it may be that at decoding, the reading of the first bit is sufficient to decode the first symbol. Both intervals are scaled as well as coded to avoid numerical accuracy problems.

In the example of Table 2, a stochastic automaton is defined for a value of T = 4. The table presents the states, transitions and outputs of a quasi-arithmetic decoder for a binary source, with T = 4 and simplification MPS / LPS.

Let E _{n be} the state of this automaton at the "bit time" n, that is, at the processing time of the received bit U _n . This state is defined by the values of the intervals [lowU _n , upU _n [and [lowS _Kn , upS _Kn [. Depending on the stationary distribution of the source (represented, column K1, by the probability of the most probable MPS symbol), the current interval [lowS _Kn , upS _Kn [will be subdivided into entire intervals. Depending on whether the input bit is U _n = 0 in the column K2 or U _n = 1 in the column K3, the subdivisions into integer intervals of the current interval [lowS _Kn , upS _Kn [possible lead to the emission of indicated MPS or LPS symbols or the appropriate rescaling before determining the next state of the PLC.

An encoder or decoder respectively using the stochastic automaton of the table 1, respectively table 2, starts in the state E _k = 1, respectively E _n = 1.

Various models are envisaged according to the invention for use in coding / decoding methods and for being implemented in the corresponding coders / decoders. Thus, it is envisaged a so-called "product" model that can be a model resulting from the product of the source and encoder models or a model resulting from the product of the source and decoder models.

In general, the states of the model produced must gather the information contained in the states of each of the two models (source and encoder or source and decoder). In the general case, the size of the state space of the "product model" can not be known a priori. It can be calculated from the binary source model and the T parameter.

For a source / encoder product model, as the number of bits in each bit produced by each transition of the produced model is random, the dependency structure between the sequence of second bits and the states of the produced model is random. In order to "capture" this dependency structure, the Markov process, represented by the X _k states of the encoder according to the product model, is augmented by a counter N _k to obtain the following dependency structure (X, N) = (X ₁ , N ₁ ) ... (X _K , N _K ). Each state X _k of the model produced is defined by the state of the encoder model, the state of the source model and the number of bits transmitted when the kth symbol has been encoded: (X _k , N _k ) = (lowS _k , upS _k , Ck, N _k ). This leads to the dependency structure graphically described by the figure 8 . According to this structure, a sequence of binary symbols S ₁ ^K is converted into a sequence of bits U ₁ ^{N K} , with N _K the number of bits transmitted when K symbols have been coded. Given a state X _k-1 and an input symbol S _k , the product model specifies the bits U _{Nk-1 + 1} to U _Nk which must be issued and the next state X _k . Some transitions may not emit any bit. The probabilities of the transitions between (X _k , N _k ) = (lowS _k , upS _k , Ck, N _k ) and the next state in the lattice (X _{k + 1} , N _{k + 1} ) = (lowS _{k + 1} , upS _{k + 1} , C _{k + 1} , N _{k + 1} ) are given by the binary source model, P (C _{k + 1} | C _k ) = P (S _{k + 1} | C _k ). The second bits Y _{Nk-1 + 1} to Y _Nk are obtained at the output of the transmission channel when the first bits U _{Nk-1 + 1} to U _Nk have been transmitted. These second bits are observations on the first bits.

An example of a product model derived from the source and encoder models is presented according to Figures 6 and 7 . The sample product model defines in the table of the Figure 6A comes from the product of the source model of the figure 4C and the encoder model of the table 1. In theory, the size of the state space resulting from this product is given by the product of the sizes of the state spaces of the two models (source and encoder). However, simplifications take place, which leads to a smaller number of states.

The arithmetic coder model of Table 1 (T = 4) using the simplified state model of column C3 was represented as a lattice Figures 7A and 7B , the Figure 7A corresponding to a probability 0.63 ≤P (MPS), the Figure 7B corresponding to a probability 0.37 ≤ P (MPS) ≤ 0.63. One or the other of these figures can be used according to the probabilities of the binary source. The probabilities of the binary source, resulting from a conversion from an M-ary source, depend on the state C _k of the model of this binary source. The transitions of the source / encoder product model are thus a function of the distribution of the source. The table of the Figure 6A is built according to the states C _k of the source model of the figure 4C , The probability of transitions S _k, S transitions _k = 0 and _k = S 1 of the figure 4C being modified in S _k = LPS and S _k = MPS, and transitions of the encoder model shown in Figures 7A and 7B depending on the probability of the source. Column M1 of the Figure 6A contains only 4 states of the product model defined in column M2 by the state variables of the source model and the corresponding encoder. The model of the source having 3 states and that of the encoder having 2 states, the model produced will have at most 6 states. In practice, simplifications take place as explained below using Figures 7A and 7B . The Figure 7A shows that for a state E _k = 0 of the encoder, it is possible to continue in a state E _k = 1 of the encoder for a probability of the source model 0.63 <P (MPS). On the other hand, Figure 7B indicates that it is not possible to reach the state E _k = 1 of the encoder from the state E _k = 0.

So in the product model of the Figure 6A No state of the product model X _k = X _k = 4 and 5 corresponding to pairs of states of the encoder and source model (E _k = 1, _k = 1 C) and (E _k = 1, C _k = 2) exists for the probability law such as 0.37 <P (MPS) <0.63 of the source model. Columns M3 and M4 indicate, for the input symbol (MPS or LPS), the transmitted bits and the next state of the produced model.

For a source / decoder model according to the FIG 9A or 9B , the states of this model produced must gather the information of the states of the source and decoder models. A state X _n of this product model is thus defined by X _n = (lowU _n , upU _n , lowS _Kn , upS _Kn , C _Kn ), K _n being the number of symbols decoded when n bits have been received. The number of symbols produced by each transition of the product model is random. As a result, the dependency structure between the second bit sequence and the decoded symbols is random. The Markov chain X ₁ ... x _N of the source / decoder product model is augmented by a counter variable K _n to give the Markov chain (X, K) = (X ₁ , K ₁ ) ... (X _N , K _N ) which leads to the dependency structure illustrated in figure 10 . According to this structure, a sequence of bits U ₁ ^N is converted into a sequence of symbols S ₁ ^{K NOT} , with K _N the number of symbols decoded when N bits have been received. Given a state X _n and an input bit U _{n + 1} , the automaton produces the sequence of binary symbols S _{Kn + 1} ^{Kn + 1} , and specifies the next state X _{n + 1} . The probabilities of transitions between (X _n , K _n ) and (X _{n + 1} , K _{n + 1} ) in the lattice depend on the binary source model. They are given by the formula in annex (17). As for the source / encoder model, simplifications of states of this model are possible depending on the model of the decoder and the source model used. In the example of the FIG 9A or 9B , the source model used is that of the figure 4C and the decoder model, that of table 2.

Robust decoding methods for arithmetic codes are based on the dependency models described above.

A first embodiment of the robust decoding method is illustrated by the figure 11 .

Reference is made to Annex 1 for the associated mathematical development.

The decoder receives as input the bitstream to be decoded. The source model, the variable N representing the number of received bits to be decoded and the probability P (U _n / Y _n ) of having transmitted a first bit (or first bit) U _n knowing the second bit (or second element binary) Y _n observed are data that the decoder has for input, that is to say that, by way of example only, this data can be received as input with the bitstream to be decoded or can be read in a memory accessible by the decoder. The probability P (U _n / Y _n ) can be obtained from the properties of the channel.

The MAP (Maximum A Posteriori) criterion corresponds to the optimal Bayesian estimation of an X process from the available observations Y as indicated by equation (3), appendix 1. Optimization is performed for all "sequences" possible. In this first embodiment, this criterion is applied to the estimation of the hidden states of the processes (X, N), which can be considered as symbol clock processes, and processes (X, K), which can be considered as bit clock process, given the bit sequences observed. The estimate can be made either with bit clock or symbol clock. For reasons of simplicity of implementation, the bit clock estimate is preferred.

In this case, the estimation of the set of hidden states (X, K) = (X ₁ , K ₁ ) ... (X _N , K _N ) is equivalent to the estimation of the corresponding decoded symbol sequence S = S ₁ ... S _Kn ... S _KN , knowing the observations Y ₁ ^N at the output of the channel. The best sequence (X, K) can be obtained from the local probabilities on the pairs (X _n , K _n ) thanks to the equation (4).

• [7] L.R. Bahl, J. Cocke, F. Jelinek, and J.Raviv. Optimal decoding of linear codes for minimizing symbol error rate. IEEE trans. on information theory, 02:284-287, March 1974 .

The calculation of P (X _n , K _n | Y) can be performed from the decomposition equation (5) where α denotes a normalization factor. Dependencies of the Markov chain allow a recursive calculation of the two terms of the right part of this equation, based on the BCJR algorithm described in the book

• [7] LR Bahl, J. Cocke, F. Jelinek, and J.Raviv. Optimal decoding of linear codes for minimizing symbol error rate. IEEE trans. on information theory, 02: 284-287, March 1974 .

Thus, in step 300, the trellis is first constructed using the source / decoder model of the Figure 9A for example, and the corresponding probabilities are calculated for each transition of the trellis. Thus, from the produced model, a forward pass allows the successive construction of the connected states by transitions of the state diagram produced from a state of origin to a final state. Each state defines an expected number of decoded symbols for a given number of bits passed from the stream of second bits from the original state. During this same step, the forward pass of the BCJR algorithm makes it possible to calculate the term P (X _n = x _n , K _n = k _n | Y ₁ ⁿ ) of the decomposition as the sum over the set of possible values of x _n-1 , k _n-1 of the product of three conditional probabilities (equation (6): the recursive probability, the transition probability between states (x _n-1 , k _n-1 ) and (x _n , k _n ) given by the model produced (as indicated in equation 17 and representing the product of the transition probabilities P (S _k | C _k-1 ) previously in the source model) and the probability of having emitted the bit U _n (which can be 0 or 1) triggering the transition between the states (x _n-1 , k _n-1 ) and (x _n , k _n ) knowing the second binary element Y _n.The latter probability is obtained from the properties of the channel. recursive calculation is initialized to the starting state (0,0) and makes it possible to calculate P (X _n , K _n | Y ₁ ⁿ ) for any possible state (x _n , k _n ) with n = 1 ... N bit clock A insi, this pass before allows the construction of the lattice of the model produced and the computation of the probabilities P (X _n , K _n | Y ₁ ⁿ ), successive and for each instant n, to end in a given state (X _n , K _n ) knowing bits of the stream of second bits Y ₁ ⁿ .

In step 302, the back pass makes it possible to obtain the term P (Y ^N _{n + 1} | X _n = x _n , K _n = k _n ) from the decomposition for any possible state (x _n , k _n ) at each bit bit n consecutively from N to 1. This term is calculated as the sum of the product of three conditional probabilities (equation (7)): the initialized recursive probability for all possible "last states" (X _N , K _N ), the transition probability between the states (X _{n + 1} , K _{n + 1} ) and (X _n , K _n ) given by the model produced and the probability of having emitted the bit U _{n + 1} triggering the transition between the states (X _{n + 1} , K _{n + 1} ) and (X _n , K _n ) knowing the second binary element Y _{n + 1} . As before, this latter probability is obtained from the channel model. Thus, this backward pass makes it possible to calculate the probabilities P (Y ^N _{n + 1} | X _n = x _N , K _n = k _n ), successive and for each instant n from N to 1, to observe bits of the flow of second bits Y ₁ ⁿ knowing a given state (X _n , K _n ).

When the number of symbols K is known, a termination constraint can be added during the backward pass in step 302. Thus, all the paths in the trellis that do not lead to the correct number of symbols K _N = K are deleted. . Thus, the backward pass comprises the successive deletion of the states of the state diagram produced, from the final state to the original state, for which the expected number of symbols for a given number of bits of the stream of second bits is different from a maximum known number of symbols. Thus, the first pass of step 300 starts at the initial state of the trellis and successively builds the different sections of the trellis. During this pass before, it is not possible to know which are the paths which do not respect the constraint of termination. Once the forward pass is complete, the states of the last section (n = N) of the trellis that do not respect the constraint are removed. The back pass then allows you to delete all branches that do not succeed.

In step 304, the probabilities obtained from the forward and backward passes are combined in the form of a product for each value of n varying from 1 to N to obtain the probabilities P (X _n , K _n | Y) for each state. not. In this way are estimated the probabilities related to the transitions of the constructed lattice.

In step 306, the product of equation (4) is performed on the probabilities P (X _n , K _n | Y ₁ ⁿ ) with n varying from 1 to N. This product is carried out for the various possible combinations of these probabilities. Thus, the highest probability P (X, K | Y) is retained and the probabilities P (X _n , K _n | Y) constituting this product make it possible to determine the most likely path in the trellis. From the lattice and the corresponding table of the source / decoder product model, it is possible to determine the symbols corresponding to this path and to reconstruct a sequence of symbols S ₁ ^K.

In another possible embodiment, the lattice on which the estimate is made may be pre-calculated and stored from the source / decoder product model and the length N of the observed bit sequence. So, with reference to the figure 12 from the source / decoder model and the values of K and N, the lattice is constructed in step 400 in a forward pass and simplified by a backward pass at step 402. This lattice is stored in memory for be used at each bit stream arrival for which the source / decoder model and the K and N values are the same.

For a received bit stream comprising N = 6 bits, for a known number of symbols to be found K = 7 and for a source / decoder model illustrated in FIG. Figure 9A , the construction of the trellis is illustrated in figure 13 . This trellis comprises, at each state (X _n , K _n ), several couples values (X _n , K _n ). For example, the pair (X ₁ , K ₁ ) can take the values (1, 1) or (2, 1) depending on whether the transmitted bit U ₁ is 0 or 1.

At each arrival of bitstream (having the same model, the same values of K and N as the trellis), a forward pass and a backward pass are made on the bit stream at steps 404 and 406, for example in a parallel manner. . The probabilities P (X _n , K _n | Y ₁ ⁿ ) and P (Y _{n + 1} ^N | X _n , K _n ) are calculated as indicated in equations (6) and (7). The product of these probabilities is carried out at step 408 for each value of n and corresponds to each transition of the trellis. It is then proceeded to calculate the product of the probabilities for n varying from 1 to N in step 410 as indicated in (4) to obtain probabilities P (X, K | Y). The maximum of these probabilities P (X, K | Y) is kept to determine the most likely path on the lattice and the corresponding symbol sequence.

According to the example of the figure 13 , the most likely path (obtaining the highest probability P (X, K | Y)) is marked in bold. The sequence of states (X ₀ , K ₀ ), ..., (X _N , K _N ) makes it possible, on reading table 8A, to determine the sequence of symbols corresponding to this path.

An optimal decoding method of quasi-arithmetic codes has been presented. This process requires the creation and manipulation of an estimation lattice, such as that of the figure 13 . The size of this lattice depends on the number of states of the source / decoder product model (4 states are possible for the product model source / decoder of the Figure 9A for example) and the amplitude of the values that the variable K _n can take at each bit bit n. The number of states of the source / decoder product model is itself dependent on the binary source model and the precision parameter of the quasi-arithmetic coding T. To adjust the compromise between the compression efficiency of the quasi-arithmetic encoder and the complexity of the estimation process, it is therefore sufficient to adjust the value of the parameter T. T value as small as 4, for example, results in a reasonable excess flow. If, however, maximum compression is required, T must have a much larger value and the minimum value of T required depends on the distribution of the source. In most of these cases, the value of T is too great to use the optimal estimation method described above. The number of states of the source / decoder product model may also be too large to pre-calculate these states.

An estimation method applicable to arithmetic coding and quasi-arithmetic coding is now described regardless of the precision T selected, that is to say whatever the compression efficiency chosen.

This process relies on the same source, encoder and decoder models as those previously seen. Since the number of states of the encoder and decoder models is too large for them to be pre-calculated, the corresponding estimation lattice can not be built either. The estimation method proposed below is therefore based on the exploration of a subset of the estimation mesh, which is considered as a tree. This process is called a sequential estimation method. It can be applied from the source / encoder models and perform the sequential calculations using the "symbol clock", that is to say a counter incremented according to the number of symbols to be found. It can also be applied from the source / decoder models and perform the sequential calculations using "bit clock", that is to say a counter incremented according to the number of bits to be decoded. This second estimation method is presented for a binary source by way of example only. It can also apply to an M-ary source (M> 2).

The figure 14 illustrates the so-called symbol clock sequential estimation method, ie based on the source / encoder model. So, according to the figure 8 representing the dependence structure of the source / encoder model, the estimation of the sequence of states (X, N) for a bit stream is equivalent to the estimation of the sequence of transitions between these states, ie to the estimation of the sequence S knowing the observations Y at the output of the channel. This results in a mathematical formulation presented in (8) where α denotes a normalization factor. In the case of optimal arithmetic coding (for a high T value), as the transmitted bits are distributed uniformly and independently, this normalization factor is P (U ₁ ^N ) / P (Y ₁ ^N ) = 1.

In step 500, the probability of decoding a sequence of symbols (from 1 to k) knowing bits of the stream of second bits P (S ₁ ^k / Y ₁ ^Nk ) for each symbol instant k can be calculated in one pass before unique thanks to the decomposition of the equation (9) where N _k is the total number of bits transmitted when in the state X _k . Taking into account the binary source model defined in equation (1) and considering a channel without memory, the probability of observing the second bits corresponding to a state of the model depends only on the first bits for this state, this probability can be rewritten as shown in equation (10).

The calculation is initialized as indicated in (11) with S ₁ = s _{1 being} able to take two possible values 0 or 1 in the case of a binary source, and M possible values in the case of an M-ary source. C ₁ is the starting state of the source model. For a channel considered without memory, the second term of this equation modeling the noise introduced by this channel is given to equation (12) as the product, for a given state X _k of the source / encoder model, probabilities of observing a second bit Y _n for a first transmitted bit U _n . Starting from this initialization, equation (10) is used to calculate P (S ₁ ^k / Y ₁ ^Nk ) for each consecutive symbol time k.

Since all the possible symbol sequences are considered, the size of the estimation mesh grows exponentially with the number of symbols K. Thus, step 502 allows the calculation of the probabilities P (S ₁ ^k / Y ₁ ^Nk ) for a number of symbols k varying from 1 to x1. For symbols from this value x1, pruning occurs at step 504 to maintain a complexity of the estimation mesh at a desired level, x1 is an integer less than or equal to the number of K symbols.

The pruning of step 504 may comprise different embodiments. By way of example, two embodiments are presented below. Achievements can be combined or used separately.

Thus, in a first embodiment of the pruning of step 504, only the most likely W nodes of a state (X _k , N _k ) are selected and this from a number k = x1 to guarantee the significance of calculated probabilities (called likelihoods). The value of W depends on the available memory resources. Thus, for each state (X _k , N _k ), only the W greatest probabilities of the calculated probabilities of decoding a sequence of symbols (from 1 to k) knowing bits of the second bit stream P (S ₁ ^k). / Y ₁ ^Nk ) are selected and stored. In a variant, it is possible to guarantee a correct termination of the estimate by refining this constraint. Thus, only the most likely nodes w will be selected and kept for each possible value of N _k (corresponding to the symbol clock). N _k is the number of bits transmitted for k coded symbols during coding. The relation between W and w is given in (19) with N _kmax and N _kmin depending respectively on the probability of the most and least likely sequence S ₁ ^k .

In a second embodiment of the pruning of step 504, the pruning criterion is based on a threshold applied to the likelihood of the paths. The likelihood of a path is given by equation (8) (symbol clock). In this equation, the terms P (Y | U) are related to the channel model as shown in equation (12). P (Y _n = y _n | U _n = u _n ) represents the probability that a bit y _n is observed knowing that a bit u _n has been transmitted or the probability of receiving a bit y _n knowing that a bit u _n was issued, which is equivalent to the probability that the observed bit is erroneous. Let p be the average bit error probability which is arbitrarily set as tolerable on the channel during the estimation. Pruning consists of deleting all paths for which the average binary error probability at the output of the channel is greater than p. Thus, at each symbol moment N _k , the pruning criterion comprises a threshold p compared to the probabilities P (Y _Nk | U _Nk ) so that all the paths of the estimation lattice for which P (Y _Nk) is verified. | U _Nk ) <(1-p) ^Nk , be deleted. Only the probabilities of decoding a sequence of symbols (from 1 to k) knowing bits of the stream of second bits P (S ₁ ^k / Y ₁ ^Nk ) calculated from the probabilities P (Y _Nk | U _Nk ) are stored. larger than the minimum threshold p.

After incrementing the value k by one unit, the probability P (S ₁ ^k / Y ₁ ^Nk ) for the number of symbols Nk (with k> x) is calculated at step 506. The pruning performed at the step 504 is reiterated for this new symbol moment Nk. Steps 504 and 506 are performed for a value k incremented up to the value K. At step 510, once the value K is reached, the posterior marginal probabilities P (S ₁ ^K / Y ₁ ^NK ) are known for all the possible sequences s ₁ ^K. The most likely sequence s ₁ ^K is chosen, that is to say the sequence for which the probability P (S ₁ ^K / Y ₁ ^NK ) is the largest among the possible sequences of the obtained estimation lattice.

When the number of bits N is known, it can constitute a constraint on the termination of the estimation lattice and the paths which do not respect the constraint N are not considered.

The figure 15 illustrates the so-called bit clock sequential estimation method, ie based on the source / decoder model. So, according to the figure 10 representing the dependence structure of the source / decoder model, the estimation of the sequence of states (X, K) to obtain a train of symbols is equivalent to the estimation of the sequence of transitions between these states, that is to say the estimation of the sequence S knowing the observations Y at the output of the channel. This results in a mathematical formulation presented in (13) where α denotes a normalization factor. In the case of optimal arithmetic coding (for a high T value), the transmitted bits being distributed uniformly and independently, this normalization factor is P (U ₁ ⁿ ) / P (Y ₁ ⁿ ) = 1.

In step 600, the probability of decoding a sequence of K _n symbols knowing the bits of the second bit stream P (S ₁ ^Kn / Y ₁ ⁿ ) for each bit n (defined later) can be calculated in a pass before unique thanks to the decomposition of the equation (13) where K _n is the total number of decoded symbols on receipt of the bit n in the state X _n . Taking into account the binary source model which defines in (2) the transition probabilities between states (X _n , K _n ) and (X _{n + 1} , K _{n + 1} ), the probability P (S ₁ ^Kn ) of obtaining a symbol sequence S ₁ ^Kn is recurrently expressed according to equation (14) using equation (15). Considering a channel without memory, that is to say that the probability of observing the second bits corresponding to a state of the model depends only on the first bits transmitted for this state, the probability P (Y ₁ ⁿ / U ₁ ⁿ ) can be rewritten and makes it possible to obtain the probability P (S ₁ ^Kn / Y ₁ ⁿ ) as indicated in (16).

As in the case of figure 14 The calculation of the probability P (S ₁ ^Kn / Y ⁿ ₁₎ by equation (16) is initialized to n = 1, S ₁ = s ₁ may take two possible values 0 or 1 in the case of a source binary, and M possible values in the case of an M-ary source. C ₁ is the starting state of the source model. Starting from this initialization, equation (16) is used to calculate P (S ₁ ^Kn / Y ₁ ⁿ ) for each consecutive bit bit n.

Since all possible symbol sequences are considered, the size of the estimation mesh grows exponentially with the number of K symbols that can be estimated. Thus, step 602 allows the calculation of probabilities P (S ₁ ^Kn / Y ₁ ⁿ ) in step 600 for a number of bits n varying from 1 to x 2, where x 2 is an integer denoting a threshold number of bits. For bits from this value x2, pruning occurs at step 604 to maintain a complexity of the estimation mesh at a desired level.

The pruning of step 604 may comprise different embodiments corresponding to the achievements of step 504 of the figure 14 . Achievements can be combined or used separately.

Thus, in a first embodiment of the pruning of step 604, only the most likely W nodes of a state (X _n , K _n ) are kept and this from a number n = x 2 to guarantee the significance of calculated probabilities (called likelihoods). Thus, for each state (X _n , K _n ), only the W greatest probabilities among the calculated probabilities of decoding a sequence of K _n symbols knowing bits of the second bit stream P (S ₁ ^k / Y ₁ ^Nk ) are stored.

In a variant, it is possible to keep only the most likely nodes w for each possible value of K _n (corresponding to the bit clock). K _n is the number of symbols transmitted for n bits received during decoding. The relation between W and w is given in (20) with K _nmax and K _nmin depending respectively on the probability of the most and least likely sequence S ₁ ^k .

In a second embodiment of the pruning of step 604, the pruning criterion is based on a threshold applied to the likelihood of the paths. The likelihood of a path is given by equation (13) (bit clock). In this equation, the terms P (Y | U) are related to the channel model as shown in Equation (16). As proposed for the figure 13 pruning consists of deleting all paths for which the average binary error probability at the output of the channel is greater than p. Thus, at each moment bit n, the pruning criterion comprises a threshold p compared to the probabilities P (Yn | Un) so that all the paths of the estimation lattice for which P (Y _n | U _n) is verified. ) <(1-p) ⁿ , be deleted. Only the probabilities of decoding a next symbol knowing bits of the second bit stream P (S ₁ ^Kn / Y ₁ ⁿ ), calculated from the probabilities P (Y _n | U _n ) greater than the minimum threshold p, are stored. . Thus, at each instant bit n, all the paths of the estimation lattice for which the pruning criterion P (Y _n | U _n ) <(1-p) ⁿ is satisfied, are deleted.

After incrementing the value n by one unit, the probability P (S ₁ ^Kn / Y ₁ ⁿ ) for the number of bits n (with n> x 2) is calculated in step 606. The pruning carried out at step 604 is taken for this new bit bit K _n . The steps 604 and 606 are performed for an incremented value n up to the value N. In step 610, once the value N is reached, the most probable sequence S ₁ ^N is chosen, that is to say the sequence for which the probability P (S ₁ ^K / Y ₁ ^N ) is the largest of the possible sequences of the obtained estimation lattice.

At the end of the estimation, the posterior marginal P (S ₁ ^K / Y ₁ ^N ) probabilities are known for all the ^K ₁ sequences remaining after pruning.

The termination constraints mentioned can be seen as a means of forcing the synchronization at the end of the symbol sequence. Indeed, they force the decoder to obtain the correct correspondence between the number of decoded symbols and the number of bits observed. However, these constraints have no synchronization effect on the middle of the sequence.

The addition to the complementary information symbol sequence makes it possible to promote re-synchronization over the entire sequence. Thus, additional symbols, or markers, are introduced at known positions in the symbol sequence. The choice of these symbols is arbitrary, and is based on the principle of the techniques described in (2, 4, 11, 5, 14). By way of example only, one of the symbols in the alphabet may be chosen or the last coded symbol may be repeated. These markers are inserted at known position according to the symbol clock. Therefore, the position of the corresponding additional bits in the bitstream depends on the coded symbol sequence, and is random.

Of course, the previously developed methods must take into account this additional information. This prior knowledge can be exploited by estimation methods. The variable K _n (k with symbol clock) indicates when a marker is expected. The corresponding transition probabilities in the estimation lattice are updated accordingly. A zero probability is assigned to all transitions that do not emit the expected markers, while a probability of 1 is assigned to the others. As a result, some paths in the trellis may be removed, resulting in a reduction in the number of states and a better ability to re-synchronize.

Examples of robust decoding methods are described above by way of example. Other variants can be envisaged. In addition, other pruning criteria may be provided and combined with the methods of the invention. M-ary / binary conversion is not required, but allows the use of small T values. The decoding methods presented above for binary sources can also work for M-ary sources.

As developed in the above description, the invention provides a method for robust decoding of arithmetic codes based on the use of arithmetic code type codes including reduced precision arithmetic codes, referred to as quasi-arithmetic codes. The precision of these quasi-arithmetic codes is an adjustable parameter depending on the desired effect. Very high accuracy achieves compression performance identical to arithmetic codes. Lower accuracy increases robustness at the cost of a reasonable loss of compression.

Compared to a so-called "instantaneous" conventional decoding, the invention makes it possible to greatly reduce the desynchronization phenomenon of the decoder. Compared to other conventional methods of so-called "sequential" decoding or on tree structures, the invention offers a greater flexibility to adjust a compromise between compression efficiency / robustness / complexity of decoding. Indeed, the invention comprises a set of methods that can be chosen according to the complexity allowed by the terminal, according to the error rate observed on the channel. A sequential decoding method includes a synchronization mechanism coupled with a pruning technique that controls the complexity of the decoding process. A trellis decoding method with parameterization of the complexity / robustness / compression compromise makes it possible to use the method in a manner coupled with a channel decoder (convolutional code decoder), which is difficult to use with traditional sequential methods.

In the proposed invention, it must be clear that the mathematical elements described are not of interest in themselves, but only insofar as they concern information flows (for example, sound signal, video ), subject to material constraints.

1 Appendix 1

$$P\left({\mathrm{VS}}_{k+1}|{\mathrm{VS}}_k\right)=P\left(S_{k+1}|{\mathrm{VS}}_k\right)$$

$${\displaystyle \underset{k=K_{\mathrm{not}}+1}{\overset{K_{\mathrm{not}+1}}{\Pi }}\mathbb{P}\left(S_k|{\mathrm{VS}}_{k-1}\right)}$$

$$\widehat{X}=\arg \underset{x}{\max }\mathbb{P}\left(X=x|Y\right).$$

$$\mathbb{P}\left(X,K|Y\right)={\displaystyle \underset{\mathrm{not}=1}{\overset{\mathrm{NOT}}{\Pi }}\mathbb{P}\left(X_{\mathrm{not}},K_{\mathrm{not}}|Y\right).}$$

$$\begin{array}{l}\mathbb{P}\left(X_{\mathrm{not}},K_{\mathrm{not}}|Y\right)=\frac{\mathbb{P}\left(X_{\mathrm{not}},K_{\mathrm{not}}|{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1^{\mathrm{not}}\right)\cdot \mathbb{P}\left(Y_{\mathrm{not}+1}^{\mathrm{NOT}}|X_{\mathrm{not}},K_{\mathrm{not}}\right)}{\mathbb{P}\left(Y_{\mathrm{not}+1}^{\mathrm{NOT}}\right)},\\ \alpha \mathbb{P}\left(X_{\mathrm{not}},K_{\mathrm{not}}|{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1^{\mathrm{not}}\right)\cdot \mathbb{P}\left(Y_{\mathrm{not}+1}^{\mathrm{NOT}}|X_{\mathrm{not}},K_{\mathrm{not}}\right),\end{array}$$

$$\begin{array}{ll}\mathbb{P}\left(X_{\mathrm{not}}=x_{\mathrm{not}},K_{\mathrm{not}}=k_{\mathrm{not}}|{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1^{\mathrm{not}}\right)={\displaystyle \underset{\left(x_{\mathrm{not}}{}_{-1},k_{\mathrm{not}-1}\right)}{\Sigma }}& \mathbb{P}\left(X_{\mathrm{not}-1}=x_{\mathrm{not}-1},K_{\mathrm{not}-1}=k_{\mathrm{not}-1}|{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1^{\mathrm{not}-1}\right)\cdot \\ & \mathbb{P}\left(X_{\mathrm{not}}=x_{\mathrm{not}},K_{\mathrm{not}}=k_{\mathrm{not}}|X_{\mathrm{not}-1}=x_{\mathrm{not}-1},K_{\mathrm{not}-1}=k_{\mathrm{not}-1}\right)\cdot \\ & \mathbb{P}\left(U_{\mathrm{not}}=u_{\left(x_{\mathrm{not}-1},k_{\mathrm{not}-1}\right)\left(x_{\mathrm{not}},k_{\mathrm{not}}\right)}|Y_{\mathrm{not}}\right).\end{array}$$

$$\begin{array}{ll}\mathbb{P}\left(X_{\mathrm{not}+1}^{\mathrm{NOT}}|X_{\mathrm{not}}=x_{\mathrm{not}},K_{\mathrm{not}}=k_{\mathrm{not}}\right)={\displaystyle \underset{\left(x_{\mathrm{not}}{}_{+1},k_{\mathrm{not}+1}\right)}{\Sigma }}& \mathbb{P}\left(Y_{\mathrm{not}+2}^{\mathrm{NOT}}|X_{\mathrm{not}+1}=x_{\mathrm{not}+1},K_{\mathrm{not}+1}=k_{\mathrm{not}+1}\right).\\ & \mathbb{P}\left(X_{\mathrm{not}+1}=x_{\mathrm{not}+1},K_{\mathrm{not}+1}=k_{\mathrm{not}+1}|X_{\mathrm{not}}=x_{\mathrm{not}},K_{\mathrm{not}}=k_{\mathrm{not}}\right)\cdot \\ & \mathbb{P}\left(U_{\mathrm{not}+1}=u_{\left(x_{\mathrm{not}},k_{\mathrm{not}}\right)\left(x_{\mathrm{not}+1},k_{\mathrm{not}+1}\right)}|Y_{\mathrm{not}+1}\right).\end{array}$$

$$\begin{array}{ll}\mathbb{P}\left(S_1^K|Y_1^{\mathrm{NOT}}\right)& =\mathbb{P}\left(S_1^K\right)\cdot \mathbb{P}\left(U_1^{\mathrm{NOT}}|Y_1^{\mathrm{NOT}}\right)\\ & \alpha \mathbb{P}\left(S_1^K\right)\cdot \mathbb{P}\left(Y_1^{\mathrm{NOT}}|U_1^{\mathrm{NOT}}\right),\end{array}$$

$$\begin{array}{ll}\mathbb{P}\left(S_1^k|Y_1^{{\mathrm{NOT}}_k}\right)\alpha & \mathbb{P}\left(S_1^{k-1}|Y_1^{{\mathrm{NOT}}_{k-1}}\right)\cdot \mathbb{P}\left(S_k|S_{k-1}\right)\cdot \\ & \mathbb{P}\left(Y_{{\mathrm{NOT}}_{k-1}+1}^{{\mathrm{NOT}}_k}|Y_1^{{\mathrm{NOT}}_{k-1}},U_1^{{\mathrm{NOT}}_k}\right),\end{array}$$

$$\begin{array}{ll}\mathbb{P}\left(S_1^k|Y_1^{{\mathrm{NOT}}_k}\right)\alpha & \mathbb{P}\left(S_1^{k-1}|Y_1^{{\mathrm{NOT}}_{k-1}}\right)\cdot \mathbb{P}\left(S_k|{\mathrm{VS}}_{k-1}\right)\cdot \\ & \mathbb{P}\left(Y_{{\mathrm{NOT}}_{k-1}+1}^{{\mathrm{NOT}}_k}|U_{{\mathrm{NOT}}_{k-1}+1}^{{\mathrm{NOT}}_k}\right)\end{array}$$

$$\begin{array}{ll}\mathbb{P}\left(S_1=s_1|Y_1^{{\mathrm{NOT}}_1={\mathrm{not}}_1}={\mathrm{there}}_1^{{\mathrm{NOT}}_1={\mathrm{not}}_1}\right)\alpha & \mathbb{P}\left(S_1=s_1|{\mathrm{VS}}_1\right)\cdot \\ & \mathbb{P}\left(Y_1^{{\mathrm{NOT}}_1}={\mathrm{there}}_1^{{\mathrm{not}}_1}|U_1^{{\mathrm{NOT}}_1}=u_1^{{\mathrm{not}}_1}\right),\end{array}$$

$$\begin{array}{l}\mathbb{P}\left(Y_{{\mathrm{NOT}}_{k-1}+1}^{{\mathrm{NOT}}_k}={\mathrm{there}}_{{\mathrm{not}}_{k-1}+1}^{{\mathrm{not}}_k}|U_{{\mathrm{NOT}}_{k-1}+1}^{{\mathrm{NOT}}_k}=u_{{\mathrm{not}}_{k-1}+1}^{{\mathrm{not}}_k}\right)=\\ |\begin{array}{l}{\displaystyle \underset{\mathrm{not}={\mathrm{NOT}}_{k-1}+1}{\overset{{\mathrm{NOT}}_k}{\Pi }}\mathbb{P}\left(Y_{\mathrm{not}}={\mathrm{there}}_{\mathrm{not}}|U_{\mathrm{not}}=u_{\mathrm{not}}\right)\mathrm{}\mathbf{yew}\mathrm{}{\mathrm{NOT}}_k>{\mathrm{NOT}}_{k-1}}\\ 1\mathrm{}\mathbf{if\; not}.\end{array}\end{array}$$

$$\mathbb{P}\left(S_1^{K_{\mathrm{not}}}|{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1^{\mathrm{not}}\right)\alpha \mathbb{P}\left(S_1^{K_{\mathrm{not}}}\right)\cdot \mathrm{<figure-callout\; id="1"\; label="P\; Y"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">\mathbb{P}\; </figure-callout>}\left(Y_{}^{}\right)\left({}_1^{\mathrm{not}}|{\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">U</figure-callout>}}_1^{\mathrm{not}}\right)$$

$$\mathbb{P}\left(S_1^{K_{\mathrm{not}}}\right)=\mathbb{P}\left(S_1^{K_{\mathrm{not}-1}}\right)\cdot \mathbb{P}\left(S_{K_{\mathrm{not}-1}+1}^{K_{\mathrm{not}}}|S_1^{K_{\mathrm{not}-1}}\right),$$

$$\begin{array}{l}\mathbb{P}\left(S_{K_{\mathrm{not}-1}+1}^{K_{\mathrm{not}}}|S_1^{K_{\mathrm{not}-1}}\right)=\\ |\begin{array}{l}{\displaystyle \underset{k=K_{\mathrm{not}-1}+1}{\overset{K_{\mathrm{not}}}{\Pi }}\mathbb{P}\left(S_{\mathrm{not}}|{\mathrm{VS}}_{k-1}\right)\mathrm{}\mathbf{if}\mathrm{}{K}_{\mathrm{not}}>K_{\mathrm{not}-1}}\\ 1\mathrm{}\mathbf{if\; not}.\end{array}\end{array}$$

$$\begin{array}{lll}\mathbb{P}\left(S_1^{K_{\mathrm{not}}}|{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1^{\mathrm{not}}\right)& \alpha & \mathbb{P}\left(S_1^{K_{\mathrm{not}-1}}\right)<figure-callout\; id="1"\; label="\cdot \; P\; Y"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">\cdot \; </figure-callout>\mathbb{P}\left(Y_{}^{}\right)\left({}_1^{\mathrm{not}-1}|{\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">U</figure-callout>}}_1^{\mathrm{not}-1}\right).\\ & & \mathbb{P}\left(Y_{\mathrm{not}}|U_{\mathrm{not}}\right)\cdot {\displaystyle \underset{k=K_{\mathrm{not}-1}+1}{\overset{K_{\mathrm{not}}}{\Pi }}\mathbb{P}\left(S_k|{\mathrm{VS}}_{k-1}\right)}\\ & \alpha & \mathbb{P}\left(S_1^{K_{\mathrm{not}-1}}|{\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1^{\mathrm{not}-1}\right)\cdot \mathbb{P}\left(Y_k|U_{\mathrm{not}}\right)\cdot {\displaystyle \underset{k=K_{\mathrm{not}-1}+1}{\overset{K_{\mathrm{not}}}{\Pi }}\mathbb{P}\left(S_k|{\mathrm{VS}}_{k-1}\right)}\end{array}$$

$$\mathbb{P}\left(X_{\mathrm{not}}=x_{\mathrm{not}},K_{\mathrm{not}}=k_{\mathrm{not}}|X_{\mathrm{not}-1}=x_{\mathrm{not}-1},K_{\mathrm{not}-1}=k_{\mathrm{not}-1}\right)={\displaystyle \underset{k=K_{\mathrm{not}-1}+1}{\overset{K_{\mathrm{not}}}{\Pi }}\mathbb{P}\left(S_k|{\mathrm{VS}}_{k-1}\right)}$$

$$\mathbb{P}\left(Y_{\mathrm{not}}|U_{\mathrm{not}}\right)<{\left(1-p\right)}^{\mathrm{not}}$$

$$W=\left({\mathrm{NOT}}_{\mathit{kmax}}-{\mathrm{NOT}}_{\mathit{kmin}}+1\right)w$$

$$W=\left(K_{\mathit{nmax}}-K_{\mathit{nmin}}+1\right)w$$

$$\begin{array}{l}\mathbb{P}\left(Y_i=0|U_i=0\right)=1-p\\ \mathbb{P}\left(Y_i=1|U_i=0\right)=p\\ \mathbb{P}\left(Y_i=0|U_i=1\right)=p\\ \mathbb{P}\left(Y_i=1|U_i=1\right)=1-p\end{array}$$

Claims (20)

1. Decoding method, in an arithmetic or almost-arithmetic decoder, binary elements received from a channel in x-ary symbols comprising the following steps:

   - receive from said channel a stream of N second binary elements Y_i corresponding to N first binary elements U_i transmitted, by means of said channel, from an arithmetic or almost-arithmetic coder to said decoder with i between 1 and N, said coder coding a source of symbols;

   - decode said N second binary elements Y_i into a stream of K x-ary symbols S_j with j between 1 and K;

   characterised in that the decoding step comprises:

   a. calculate (306, 410) probabilities of resulting in product states knowing bits of the stream of second binary elements Y_i,
   from a model of said channel,
   from a model of said source and
   from a product model defining a correspondence between said x-ary symbols S_j and said first binary elements U_i, from product states in a 'product' states diagram, a product state comprising a source state and a state of a model of said coder or of said arithmetic or almost-arithmetic decoder,

   b. determine the most probable stream (308. 412, 510, 610) of x-ary symbols from probabilities calculated in step a.
2. Method according to claim 1, characterised in that each product state comprises a source state (Ck) and a state of said model of said arithmetic or almost-arithmetic coder, said state of said model of said coder comprising: a low limit value (lowS_k) of an interval resulting from successive subdivisions of an interval corresponding to the coding of a sequence of k x-ary symbols, an upper limit value of said interval (upS_k) and a counter (N_k) representing the number of first binary elements associated with said k first x-ary symbols.
3. Method according to claim 1, characterised in that said product state comprises a source state and a state of said model of said arithmetic or almost-arithmetic decoder, said state of said model of said decoder comprising: a value of a low limit (lowU_n) of a first segment of an interval defined by the reception of a sequence of n second binary elements, a value of an upper limit (upU_n) of said first segment, a value of a low limit (lowS_Kn) of a second segment of said interval obtained when a sequence of x-ary symbols is determined from said n second binary elements, a value of an upper limit (upS_Kn) of said second interval and a counter (K_n) representing the number of determined x-ary symbols associated with said n second binary elements received.
4. Method according to claim 1, characterised in that it comprises a determination step of a stream of m-ary symbols from the stream of determined x-ary symbols,
   m being greater than or equal to x and the source model comprising transitions associated with a correspondence between source m-ary symbols (A_k) and target x-ary symbols (S_k).
5. Method according to one of the aforementioned claims, characterised in that the product and coder or decoder models are of arithmetic type.
6. Method according to one of the aforementioned claims, characterised in that step a. comprises the calculations, by steps, of decoder products an x-ary symbol knowing stream bits of second binary elements.
7. Method according to claim 6, characterised in that step a. comprises the generation of a tree constituted by states linked by transitions, each state corresponding to a probability calculation step and each transition, to the start of a state, corresponding to one of these calculated probabilities.
8. Method according to claims 1 and 7, characterised in that step b. comprises the calculation of the product of the probabilities corresponding to successive transitions in the tree and the selection of the largest product corresponding to the most probable successive transitions and to the most probable stream of x-ary symbols.
9. Method according to claims 6 and 8, characterised in that step a. comprises the calculation, for a given number of successive x-ary symbols to decode, of the probabilities of decoding a sequence of x-ary symbols knowing bits from the stream of second binary elements.
10. Method according to claims 6 and 8, characterised in that step a. comprises the calculation, for a given number of successive bits of the stream of second binary elements, of the probabilities of decoding a sequence of x-ary symbols knowing bits from the stream of second binary elements.
11. Method according to one of claims 1 to 5, characterised in that from the product model establishing said correspondence, step a. comprises

    c1. the successive construction of states linked by transitions of a diagram of states produced from an original state to a final state, each state defining an expected number of x-ary symbols decoded for a given number of bits of the stream of second binary elements from the original state.
12. Method according to claim 11, characterised in that the construction of step a. comprises

    c2. the successive deletion of states of the diagram of states produced, from the final state to the original state, for which the expected number of x-ary symbols for a given number of bits of the stream of second binary elements is different from a maximum known number of x-ary symbols.
13. Method according to one of claims 11 to 12, characterised in that step a. comprises

    c1. the calculation, after each state constructed, of the first successive probabilities to result in a given state knowing past bits of the stream of second binary elements,

    c2. the calculation, for each state from the final state to the original state, of the second successive probabilities to observe the next bits of the stream of second binary elements knowing this given state.
14. Method according to one of claims 11 to 12, characterised in that step a. comprises

    c3. the calculation, for each state of the diagram from the original state to the final state, of the first successive probabilities to result in this given state knowing the past bits of the stream of second binary elements, and, for each state from the final state to the original state, of the second successive probabilities to observe the next bits of the stream of second binary elements knowing a given status.
15. Method according to one of claims 11 to 14, characterised in that step a. comprises the calculation of the probability to result in this given state knowing the bits of the stream of second binary elements by obtaining the product of the first and second probabilities for each state.
16. Method according to claims 1 and 15, characterised in that step b. comprises the calculation of the products of probabilities of the different successive states possible then the selection of the maximum product.
17. Method according to claims 11 and 16, characterised in that, the correspondence defined by the product model being established in the form of a table, the step b. comprises the establishment of the stream of x-ary symbols corresponding to these successive states by reading said table.
18. Arithmetic or almost-arithmetic decoding device of binary elements received from a channel in x-ary symbols comprising,
    means for receiving from said channel a stream of N second binary elements Y_i corresponding to N first binary elements U_i transmitted, by means of said channel, from an arithmetic or almost-arithmetic coder to said decoder with i between 1 and N, said coder coding a source of symbols;
    means for decoding said N second binary elements Y_i into a stream of K x-ary symbols S_j with j between 1 and K;
    characterised in that it comprises means for calculating probabilities of resulting in product states knowing bits of the stream of second binary elements Y, from a model of said channel,
    from a model of said source and
    from a product model defining a correspondence between said x-ary symbols S_j and said first binary elements U_i, according to product states in
    a diagram of 'product' states, a product state comprising a source state and a state of a model of said coder or said arithmetic or almost-arithmetic decoder, and
    means to determine the most probable stream of x-ary symbols from calculated probabilities.
19. Device according to claim 18, characterised in that each product state comprises a source state (Ck) and a state of said model of said arithmetic or almost-arithmetic coder, said coder model state comprising: a low limit value (lowS_k) of an interval resulting from successive subdivisions of an interval corresponding to the coding of a sequence of k x-ary symbols, an upper limit value of said interval (upS_k) and a counter (N_k) representing the number of first binary elements associated with the k first x-ary symbols.
20. Device according to claim 18, characterised in that said product state comprises a source state and a state of said model of said arithmetic or almost-arithmetic decoder, said state of said model of said decoder comprising: a value of a low limit (lowU_n) of a first segment of an interval defined by the reception of a sequence of n second binary elements, a value of an upper limit (upU_n) of said first segment, a value of a low limit (lowSK_n) of a second segment of said interval obtained when a sequence of x-ary symbols is determined from said n second binary elements, a value of an upper limit (upS_Kn) of said second interval and a counter (K_n) representing the number of determined x-ary symbols associated with said n second binary elements received.

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